Let A be a nonzero matrix of size n. Let a k*k submatrix of A be defined as a matrix obtained by deleting any n-k rows and n-k columns of A. Let m denote the largest integer such that some m*m submatrix has a nonzero determinant. Then rank(A) = k.(adsbygoogle = window.adsbygoogle || []).push({});

Conversely suppose that rank(A) = m. There exists a m*m submatrix has a nonzero determinant.

I'm currently trying to prove this theorem. Not quite sure if I should proceed by examining the solution space of A or rather just do something clever with the determinants. I feel like there's a property of determinants that I'm missing that'd make this much easier.

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# Rank and submatrices theorem

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